1 G COORDINATE PLANES, LINES, AND LINEAR FUNCTIONS RECTANGULAR COORDINATE SYSTEMS Just as points on a coordinate line can be associated with real numbers, so points in a plane can be associated with pairs of real numbers b introducing a rectangular coordinate sstem (also called a Cartesian coordinate sstem). A rectangular coordinate sstem consists of two perpendicular coordinate lines, called coordinate aes, that intersect at their origins. Usuall, but not alwas, one ais is horizontal with its positive direction to the right, and the other is vertical with its positive direction up. The intersection of the aes is called the origin of the coordinate sstem. It is common to call the horizontal ais the -ais and the vertical ais the -ais, in which case the plane and the aes together are referred to as the -plane (Figure G.). Although labeling the aes with the letters and is common, other letters ma be more appropriate in specific applications. Figure G. shows a uv-plane and a ts-plane the first letter in the name of the plane alwas refers to the horizontal ais and the second to the vertical ais. Origin -plane -ais -ais uv-plane v u ts-plane s t Figure G. Figure G. -coordinate b P(a, b) a -coordinate Figure G. COORDINATES Ever point P in a coordinate plane can be associated with a unique ordered pair of real numbers b drawing two lines through P, one perpendicular to the -ais and the other perpendicular to the -ais (Figure G.). If the first line intersects the -ais at the point with coordinate a and the second line intersects the -ais at the point with coordinate b, then we associate the ordered pair of real numbers (a, b) with the point P. The number a is called the -coordinate or abscissa of P and the number b is called the -coordinate or ordinate of P. We will sa that P has coordinates (a, b) and write P(a,b)when we want to emphasize that the coordinates of P are (a, b). We can also reverse the above procedure and find the point P associated with the coordinates (a, b) b locating the intersection of the dashed lines in Figure G.. Because of this one-to-one correspondence between coordinates and points, we will sometimes blur the distinction between points and ordered pairs of numbers b talking about the point (a, b). G
2 G Appendi G: Coordinate Planes, Lines, and Linear Functions REMARK Recall that the smbol (a, b) also denotes the open interval between a and b; the appropriate interpretation will usuall be clear from the contet. Quadrant II (, +) Quadrant III (, ) Quadrant I (+, +) Quadrant IV (+, ) In a rectangular coordinate sstem the coordinate aes divide the rest of the plane into four regions called quadrants. These are numbered counterclockwise with roman numerals as shown in Figure G.. As indicated in that figure, it is eas to determine the quadrant in which a given point lies from the signs of its coordinates: a point with two positive coordinates (+, +) lies in Quadrant I, a point with a negative -coordinate and a positive -coordinate (, +) lies in Quadrant II, and so forth. Points with a zero -coordinate lie on the -ais and points with a zero -coordinate lie on the -ais. To plot a point P(a,b)means to locate the point with coordinates (a, b) in a coordinate plane. For eample, in Figure G. we have plotted the points P(, ), Q(, ), R(, ), and S(, ) Observe how the signs of the coordinates identif the quadrants in which the points lie. Figure G. Figure G. Q(, ) 7 7 R(, ) S(, ) P(, ) GRAPHS The correspondence between points in a plane and ordered pairs of real numbers makes it possible to visualize algebraic equations as geometric curves, and, conversel, to represent geometric curves b algebraic equations. To understand how this is done, suppose that we have an -coordinate sstem and an equation involving two variables and, sa =, =, = +, or + = We define a solution of such an equation to be an ordered pair of real numbers (a, b) whose coordinates satisf the equation when we substitute = a and = b. For eample, the ordered pair (, ) is a solution of the equation =, since the equation is satisfied b = and = (verif). However, the ordered pair (, ) is not a solution of this equation, since the equation is not satisfied b = and = (verif). The following definition makes the association between equations in and and curves in the -plane. G. definition The set of all solutions of an equation in and is called the solution set of the equation, and the set of all points in the -plane whose coordinates are members of the solution set is called the graph of the equation. One of the main themes in calculus is to identif the eact shape of a graph. Point plotting is one approach to obtaining a graph, but this method has limitations, as discussed in the following eample.
3 Appendi G: Coordinate Planes, Lines, and Linear Functions G Eample of this method. Use point plotting to sketch the graph of =. Discuss the limitations Solution. The solution set of the equation has infinitel man members, since we can substitute an arbitrar value for into the right side of = and compute the associated to obtain a point (, ) in the solution set. The fact that the solution set has infinitel man members means that we cannot obtain the entire graph of = b point plotting. However, we can obtain an approimation to the graph b plotting some sample members of the solution set and connecting them with a smooth curve, as in Figure G.. The problem with this method is that we cannot be sure how the graph behaves between the plotted points. For eample, the curves in Figure G.7 also pass through the plotted points and hence are legitimate candidates for the graph in the absence of additional information. Moreover, even if we use a graphing calculator or a computer program to generate the graph, as in Figure G.8, we have the same problem because graphing technolog uses point-plotting algorithms to generate graphs. Indeed, in Appendi A we see eamples where graphing technolog can be fooled into producing grossl inaccurate graphs. = (, ) 9 9 Figure G. (, ) (, ) (, ) (, 9) (, ) (, ) (, 9) = Figure G In spite of its limitations, point plotting b hand or with the help of graphing technolog can be useful, so here are two more eamples. Eample Sketch the graph of =. Figure G.8 [, ] [, ] Scl =, Scl = = Solution. If <, then is an imaginar number. Thus, we can onl plot points for which, since points in the -plane have real coordinates. Figure G.9 shows the graph obtained b point plotting and a graph obtained with a graphing calculator. = (, ) (, ) (, ) (, ) (,.) (, ) (,.7) (, ) = [, ] [, ] Scl =, Scl = Figure G.9 =
4 G Appendi G: Coordinate Planes, Lines, and Linear Functions Eample Sketch the graph of =. Solution. To calculate coordinates of points on the graph of an equation in and,itis desirable to have epressed in terms of or in terms of. In this case it is easier to epress in terms of, so we rewrite the equation as = Members of the solution set can be obtained from this equation b substituting arbitrar values for in the right side and computing the associated values of (Figure G.). Figure G. = 8 = (, ) 8 8 (8, ) (, ) (, ) (, ) (, ) (, ) (8, ) REMARK Most graphing calculators and computer graphing programs require that be epressed in terms of to generate a graph in the -plane. In Appendi A we discuss a method for circumventing this restriction. Eample Sketch the graph of = /. Solution. Because / is undefined at =, we can onl plot points for which =. This forces a break, called a discontinuit, in the graph at = (Figure G.). = / = / (, ),, (, ),,,, (, b) -intercept is b Figure G. (, ),, -intercept is a Figure G. (a, ) INTERCEPTS Points where a graph intersects the coordinate aes are of special interest in man problems. As illustrated in Figure G., intersections of a graph with the -ais have the form (a, ) and intersections with the -ais have the form (,b). The number a is called an -intercept of the graph and the number b a -intercept.
5 Appendi G: Coordinate Planes, Lines, and Linear Functions G Eample Find all intercepts of (a) + = (b) = (c) = / + = Solution (a). To find the -intercepts we set = and solve for : = or = To find the -intercepts we set = and solve for : = or = As we will see later, the graph of + = is the line shown in Figure G.. Figure G. Solution (b). To find the -intercepts, set = and solve for : = Thus, = is the onl -intercept. To find the -intercepts, set = and solve for : = ( ) = So the -intercepts are = and =. The graph is shown in Figure G.. Solution (c). To find the -intercepts, set = : = This equation has no solutions (wh?), so there are no -intercepts. To find -intercepts we would set = and solve for. But substituting = leads to a division b zero, which is not allowed, so there are no -intercepts either. The graph of the equation is shown in Figure G.. P (, ) P (, ) (Rise) (Run) SLOPE To obtain equations of lines we will first need to discuss the concept of slope, which is a numerical measure of the steepness of a line. Consider a particle moving left to right along a nonvertical line from a point P (, ) to a point P (, ). As shown in Figure G., the particle moves units in the -direction as it travels units in the positive -direction. The vertical change is called the rise, and the horizontal change the run. The ratio of the rise over the run can be used to measure the steepness of the line, which leads us to the following definition. Figure G. G. definition If P (, ) and P (, ) are points on a nonvertical line, then the slope m of the line is defined b m = rise run = () REMARK Observe that this definition does not appl to vertical lines. For such lines we have = (a zero run), which means that the formula for m involves a division b zero. For this reason, the slope of a vertical line is undefined, which is sometimes described informall b stating that a vertical line has infinite slope.
6 G Appendi G: Coordinate Planes, Lines, and Linear Functions P (, ) P (, ) P (, ) P (, ) When calculating the slope of a nonvertical line from Formula (), it does not matter which two points on the line ou use for the calculation, as long as the are distinct. This can be proved using Figure G. and similar triangles to show that m = = Moreover, once ou choose two points to use for the calculation, it does not matter which one ou call P and which one ou call P because reversing the points reverses the sign of both the numerator and denominator of () and hence has no effect on the ratio. Figure G. Eample In each part find the slope of the line through (a) the points (, ) and (9, 8) (b) the points (, 9) and (, ) (c) the points (, 7) and (, 7). Solution. (a) m = 8 9 = = (b) m = 9 = 7 7 = (c) m = ( ) = Eample 7 Figure G. shows the three lines determined b the points in Eample and eplains the significance of their slopes. P (9, 8) P (, 9) (, 7) (, 7) P (, ) P (, ) m = Traveling left to right, a point on the line rises two units for each unit it moves in the positive -direction. Figure G. m = Traveling left to right, a point on the line falls three units for each unit it moves in the positive -direction. m = Traveling left to right, a point on the line neither rises nor falls. As illustrated in this eample, the slope of a line can be positive, negative, or zero. A positive slope means that the line is inclined upward to the right, a negative slope means that the line is inclined downward to the right, and a zero slope means that the line is horizontal. An undefined slope means that the line is vertical. Figure G.7 shows various lines through the origin with their slopes.
7 Appendi G: Coordinate Planes, Lines, and Linear Functions G7 Figure G.7 Positive m = m = m = m = m = m = m = m = m = Negative slope slope PARALLEL AND PERPENDICULAR LINES The following theorem shows how slopes can be used to tell whether two lines are parallel or perpendicular. G. theorem (a) Two nonvertical lines with slopes m and m are parallel if and onl if the have the same slope, that is, m = m (b) Two nonvertical lines with slopes m and m are perpendicular if and onl if the product of their slopes is, that is, m m = This relationship can also be epressed as m = /m or m = /m, which states that nonvertical lines are perpendicular if and onl if their slopes are negative reciprocals of one another. A complete proof of this theorem is a little tedious, but it is not hard to motivate the results informall. Let us start with part (a). Suppose that L and L are nonvertical parallel lines with slopes m and m, respectivel. If the lines are parallel to the -ais, then m = m =, and we are done. If the are not parallel to the -ais, then both lines intersect the -ais; and for simplicit assume that the are oriented as in Figure G.8a. On each line choose the point whose run relative to the point of intersection with the -ais is. On line L the corresponding rise will be m and on L it will be m. However, because the lines are parallel, the shaded triangles in the figure must be congruent (verif), so m = m. Conversel, the condition m = m can be used to show that the shaded triangles are congruent, from which it follows that the lines make the same angle with the -ais and hence are parallel (verif). L L L Rise = m Rise = m Rise = Rise = m Run = Run = Run = /m Run = L Figure G.8 (a) (b) Now suppose that L and L are nonvertical perpendicular lines with slopes m and m, respectivel; and for simplicit assume that the are oriented as in Figure G.8b. On line L choose the point whose run relative to the point of intersection of the lines is, in which case the corresponding rise will be m ; and on line L choose the point whose rise relative to the point of intersection is, in which case the corresponding run will be /m. Because the lines are perpendicular, the shaded triangles in the figure must be congruent (verif), and hence the ratios of corresponding sides of the triangles must be equal. Taking into account that for line L the vertical side of the triangle has length and the horizontal side has length /m (since m is negative), the congruence of the triangles implies that m / = ( /m )/ orm m =. Conversel, the condition m = /m can be used
8 G8 Appendi G: Coordinate Planes, Lines, and Linear Functions to show that the shaded triangles are congruent, from which it can be deduced that the lines are perpendicular (verif). B(, 7) A(, ) Figure G.9 C(7, ) Eample 8 Use slopes to show that the points A(, ), B(, 7), and C(7, ) are vertices of a right triangle. Solution. We will show that the line through A and B is perpendicular to the line through B and C. The slopes of these lines are m = 7 = and m = 7 7 = Slope of the line through A and B Slope of the line through B and C Since m m =, the line through A and B is perpendicular to the line through B and C; thus, ABC is a right triangle (Figure G.9). L (, b) L (a, ) Ever point on L has an -coordinate of a and ever point on L has a -coordinate of b. Figure G. LINES PARALLEL TO THE COORDINATE AXES We now turn to the problem of finding equations of lines that satisf specified conditions. The simplest cases are lines parallel to the coordinate aes. A line parallel to the -ais intersects the -ais at some point (a, ). This line consists precisel of those points whose -coordinates equal a (Figure G.). Similarl, a line parallel to the -ais intersects the -ais at some point (,b). This line consists precisel of those points whose -coordinates equal b (Figure G.). Thus, we have the following theorem. G. theorem The vertical line through (a, ) and the horizontal line through (, b) are represented, respectivel, b the equations = a and = b Eample 9 The graph of = is the vertical line through (, ), and the graph of = 7 is the horizontal line through (, 7) (Figure G.). = (, 7) = 7 m Figure G. (, ) P There is a unique line through P with slope m. Figure G. LINES DETERMINED BY POINT AND SLOPE There are infinitel man lines that pass through an given point in the plane. However, if we specif the slope of the line in addition to a point on it, then the point and the slope together determine a unique line (Figure G.). Let us now consider how to find an equation of a nonvertical line L that passes through a point P (, ) and has slope m. IfP(,) is an point on L, different from P, then the slope m can be obtained from the points P(,) and P (, ); this gives m =
9 Appendi G: Coordinate Planes, Lines, and Linear Functions G9 which can be rewritten as = m( ) () With the possible eception of (, ), we have shown that ever point on L satisfies (). But =, = satisfies (), so that all points on L satisf (). We leave it as an eercise to show that ever point satisfing () lies on L. In summar, we have the following theorem. G. theorem The line passing through P (, ) and having slope m is given b the equation = m( ) () This is called the point-slope form of the line. Eample Find the point-slope form of the line through (, ) with slope. Solution. Substituting the values =, =, and m = in () ields the pointslope form + = ( ). LINES DETERMINED BY SLOPE AND -INTERCEPT A nonvertical line crosses the -ais at some point (,b). point-slope form of its equation, we obtain b = m( ) which we can rewrite as = m + b. To summarize: If we use this point in the When the equation of a line is written as in (), the slope is the coefficient of and the -intercept is the constant term (Figure G.). G. theorem The line with -intercept b and slope m is given b the equation This is called the slope-intercept form of the line. = m + b () = m + b Eample Slope Figure G. -intercept equation slope -intercept = + 7 = + = = 8 = m = m = m = m = m = b = 7 b = b = b = 8 b = Eample stated conditions: Find the slope-intercept form of the equation of the line that satisfies the (a) slope is 9; crosses the -ais at (, ) (b) slope is ; passes through the origin
10 G Appendi G: Coordinate Planes, Lines, and Linear Functions (c) passes through (, ); perpendicular to = + (d) passes through (, ) and (, ). Solution (a). = 9. From the given conditions we have m = 9 and b =, so () ields Solution (b). From the given conditions m = and the line passes through (, ), so b =. Thus, it follows from () that = + or =. Solution (c). The given line has slope, so the line to be determined will have slope m =. Substituting this slope and the given point in the point-slope form () and then simplifing ields ( ) = ( ) = + Solution (d). We will first find the point-slope form, then solve for in terms of to obtain the slope-intercept form. From the given points the slope of the line is m = = 9 We can use either of the given points for (, ) in (). We will use (, ). This ields the point-slope form = 9( ) Solving for in terms of ields the slope-intercept form = 9 We leave it for the reader to show that the same equation results if (, ) rather than (, ) is used for (, ) in (). THE GENERAL EQUATION OF A LINE An equation that is epressible in the form A + B + C = () where A, B, and C are constants and A and B are not both zero, is called a first-degree equation in and. For eample, + = is a first-degree equation in and since it has form () with A =, B =, C = In fact, all the equations of lines studied in this section are first-degree equations in and. The following theorem states that the first-degree equations in and are precisel the equations whose graphs in the -plane are straight lines. G.7 theorem Ever first-degree equation in and has a straight line as its graph and, conversel, ever straight line can be represented b a first-degree equation in and. Because of this theorem, () is sometimes called the general equation of a line or a linear equation in and.
11 Appendi G: Coordinate Planes, Lines, and Linear Functions G Eample Graph the equation + =. (, ) (, ) Solution. Since this is a linear equation in and, its graph is a straight line. Thus, to sketch the graph we need onl plot an two points on the graph and draw the line through them. It is particularl convenient to plot the points where the line crosses the coordinate aes. These points are (, ) and (, ) (verif), so the graph is the line in Figure G.. + = Figure G. Eample Find the slope of the line in Eample. Solution. Solving the equation for ields = + which is the slope-intercept form of the line. Thus, the slope is m =. LINEAR FUNCTIONS When and are related b Equation () we sa that is a linear function of. Linear functions arise in a variet of practical problems. Here is a tpical eample. Eample A universit parking lot charges $. per da but offers a $. monthl sticker with which the student pas onl $. per da. (a) Find equations for the cost C of parking for das per month under both pament methods, and graph the equations for. (Treat C as a continuous function of, even though onl assumes integer values.) (b) Find the value of for which the graphs intersect, and discuss the significance of this value. Cost in dollars C Figure G. C = C = +. Number of parking das Solution (a). The cost in dollars of parking for das at $. per da is C =, and the cost for the $. sticker plus das at $. per da is C = +. (Figure G.). Solution (b). The graphs intersect at the point where = +. which is = /.7.. This value of is not an option for the student, since must be an integer. However, it is the dividing point at which the monthl sticker method becomes less epensive than the dail pament method; that is, for it is cheaper to bu the monthl sticker and for it is cheaper to pa the dail rate. DIRECT PROPORTION A variable is said to be directl proportional to a variable if there is a positive constant k, called the constant of proportionalit, such that = k () The graph of this equation is a line through the origin whose slope k is the constant of proportionalit. Thus, linear functions are appropriate in phsical problems where one variable is directl proportional to another.
12 G Appendi G: Coordinate Planes, Lines, and Linear Functions Hooke s law in phsics provides a nice eample of direct proportion. It follows from this law that if a weight of units is suspended from a spring, then the spring will be stretched b an amount that is directl proportional to, that is, = k (Figure G.). The constant k is the reciprocal of the stiffness of the spring and is called its compliance. is directl proportional to. Figure G. Eample pounds. Figure G.7 shows an old-fashioned spring scale that is calibrated in (a) Given that the pound scale marks are. in apart, find an equation that epresses the length that the spring is stretched (in inches) in terms of the suspended weight (in pounds). (b) Graph the equation obtained in part (a). Length stretched (in) =. 8 Weight (lb) Figure G.7 Solution (a). It follows from Hooke s law that is related to b an equation of the form = k. Tofind k we rewrite this equation as k = / and use the fact that a weight of = lb stretches the spring =. in. Thus, k = =. =. and hence =. Solution (b). The graph of the equation =. is shown in Figure G.7. EXERCISE SET G. Draw the rectangle, three of whose vertices are (, ), (, ), and (, 7), and find the coordinates of the fourth verte.. Draw the triangle whose vertices are (, ), (, ), and (, ), and find its area. Draw a rectangular coordinate sstem and sketch the set of points whose coordinates (, ) satisf the given conditions.. (a) = (b) = (c) (d) = (e) (f ). (a) = (b) = (c) < (d) and (e) = (f) = Sketch the graph of the equation. (A calculating utilit will be helpful in some of these problems.) Hooke s law, named for the English phsicist Robert Hooke ( 7), applies onl for small displacements that do not stretch the spring to the point of permanentl distorting it.
13 Appendi G: Coordinate Planes, Lines, and Linear Functions G. =. = + 7. = 8. = =. =. =. =. Find the slope of the line through (a) (, ) and (, ) (b) (, ) and (7, ) (c) (, ) and (, ) (d) (, ) and (, ).. Find the slopes of the sides of the triangle with vertices (, ), (, ), and (, 7).. Use slopes to determine whether the given points lie on the same line. (a) (, ), (, ), and (, ) (b) (, ), (, ), and (, ). Draw the line through (, ) with slope (a) m = (b) m = (c) m =. 7. Draw the line through (, ) with slope (a) m = (b) m = (c) m =. 8. An equilateral triangle has one verte at the origin, another on the -ais, and the third in the first quadrant. Find the slopes of its sides. 9. List the lines in the accompaning figure in the order of increasing slope. I Figure E-9 II. List the lines in the accompaning figure in the order of increasing slope. III IV. Find and if the line through (, ) and (, ) has slope, and the line through (, ) and (7, ) has slope. 7. Use slopes to show that (, ), (, ), (, ), and (, ) are vertices of a parallelogram. 8. Use slopes to show that (, ), (, ), and (, 9) are vertices of a right triangle. 9. Graph the equations (a) + = (b) = (c) = (d) = 7.. Graph the equations (a) = (b) = 8 (c) = (d) = +.. Graph the equations (a) = (b) = (c) =.. Graph the equations (a) = (c) =. (b) =. Find the slope and -intercept of (a) = + (b) = (c) + = 8 (d) = (e) a + b =.. Find the slope and -intercept of (a) = + (b) = + (c) + = (d) = (e) a + a = (a = ). Use the graph to find the equation of the line in slopeintercept form.. I Figure E- II. A particle, initiall at (, ), moves along a line of slope m = to a new position (, ). (a) Find if =. (b) Find if =.. A particle, initiall at (7, ), moves along a line of slope m = to a new position (, ). (a) Find if = 9. (b) Find if =.. Let the point (,k) lie on the line of slope m = through (, ); find k.. Given that the point (k, ) is on the line through (, ) and (, ), find k.. Find if the slope of the line through (, ) and (, ) is the negative of the slope of the line through (, ) and (, ). III IV. (a) (a) (b) (b)
14 G Appendi G: Coordinate Planes, Lines, and Linear Functions 7 8 Find the slope-intercept form of the line satisfing the given conditions. 7. Slope =, -intercept =. 8. m =, b =. 9. The line is parallel to = and its -intercept is 7.. The line is parallel to + = and passes through (, ).. The line is perpendicular to = + 9 and its -intercept is.. The line is perpendicular to = 7 and passes through (, ).. The line passes through (, ) and (, 7).. The line passes through (, ) and (, ).. The -intercept is and the -intercept is.. The -intercept is b and the -intercept is a. 7. The line is perpendicular to the -ais and passes through (, ). 8. The line is parallel to = and passes through (, 8). 9. In each part, classif the lines as parallel, perpendicular, or neither. (a) = 7 and = + 9 (b) = and = 7 (c) + = and + 7 = (d) A + B + C = and B A + D = (e) = ( ) and 7 = ( ). In each part, classif the lines as parallel, perpendicular, or neither. (a) = + and = (b) = ( ) and = ( + 7) (c) = and + 9 = (d) A + B + C = and A + B + D = (e) = and =. For what value of k will the line + k = (a) have slope (b) have -intercept (c) pass through the point (, ) (d) be parallel to the line = (e) be perpendicular to the line + =?. Sketch the graph of = and eplain how this graph is related to the graphs of = and =.. Sketch the graph of ( )( + ) = and eplain how it is related to the graphs of = and + =.. Graph F = 9 C + in a CF-coordinate sstem.. Graph u = v in a uv-coordinate sstem.. Graph Y = X + inayx-coordinate sstem. 7. A point moves in the -plane in such a wa that at an time t its coordinates are given b = t + and = t. B epressing in terms of, show that the point moves along a straight line. 8. A point moves in the -plane in such a wa that at an time t its coordinates are given b = + t and = t. B epressing in terms of, show that the point moves along a straight-line path and specif the values of for which the equation is valid. 9. Find the area of the triangle formed b the coordinate aes and the line through (, ) and (, ).. Draw the graph of 9 =.. A spring with a natural length of in stretches to a length of in when a lb object is suspended from it. (a) Use Hooke s law to find an equation that epresses the amount b which the spring is stretched (in inches) in terms of the suspended weight (in pounds). (b) Graph the equation obtained in part (a). (c) Find the length of the spring when a lb object is suspended from it. (d) What is the largest weight that can be suspended from the spring if the spring cannot be stretched to more than twice its natural length?. The spring in a heav-dut shock absorber has a natural length of ft and is compressed. ft b a load of ton. An additional load of tons compresses the spring an additional ft. (a) Assuming that Hooke s law applies to compression as well as etension, find an equation that epresses the length that the spring is compressed from its natural length (in feet) in terms of the load (in tons). (b) Graph the equation obtained in part (a). (c) Find the amount that the spring is compressed from its natural length b a load of tons. (d) Find the maimum load that can be applied if safet regulations prohibit compressing the spring to less than half its natural length. Confirm that a linear function is appropriate for the relationship between and. Find a linear equation relating and, and verif that the data points lie on the graph of our equation There are two common sstems for measuring temperature, Celsius and Fahrenheit. Water freezes at Celsius ( C) and Fahrenheit ( F); it boils at C and F. (a) Assuming that the Celsius temperature T C and the Fahrenheit temperature T F are related b a linear equation, find the equation. (cont.)
15 Appendi G: Coordinate Planes, Lines, and Linear Functions G (b) What is the slope of the line relating T F and T C if T F is plotted on the horizontal ais? (c) At what temperature is the Fahrenheit reading equal to the Celsius reading? (d) Normal bod temperature is 98. F. What is it in C?. Thermometers are calibrated using the so-called triple point of water, which is 7. K on the Kelvin scale and. C on the Celsius scale. A one-degree difference on the Celsius scale is the same as a one-degree difference on the Kelvin scale, so there is a linear relationship between the temperature T C in degrees Celsius and the temperature T K in kelvins. (a) Find an equation that relates T C and T K. (b) Absolute zero ( K on the Kelvin scale) is the temperature below which a bod s temperature cannot be lowered. Epress absolute zero in C. 7. To the etent that water can be assumed to be incompressible, the pressure p in a bod of water varies linearl with the distance h below the surface. (a) Given that the pressure is atmosphere ( atm) at the surface and.9 atm at a depth of m, find an equation that relates pressure to depth. (b) At what depth is the pressure twice that at the surface? 8. A resistance thermometer is a device that determines temperature b measuring the resistance of a fine wire whose resistance varies with temperature. Suppose that the resistance R in ohms ( ) varies linearl with the temperature T in C and that R =. when T = C and that R =.9 when T = C. (a) Find an equation for R in terms of T. (b) If R is measured eperimentall as 8., what is the temperature? 9. Suppose that the mass of a spherical mothball decreases with time, due to evaporation, at a rate that is proportional to its surface area. Assuming that it alwas retains the shape of a sphere, it can be shown that the radius r of the sphere decreases linearl with the time t. (a) If, at a certain instant, the radius is.8 mm and das later it is.7 mm, find an equation for r (in millimeters) in terms of the elapsed time t (in das). (b) How long will it take for the mothball to completel evaporate? 7. The accompaning figure shows three masses suspended from a spring: a mass of g, a mass of g, and an unknown mass of W g. (a) What will the pointer indicate on the scale if no mass is suspended? (b) Find W. g mm mm mm Figure E-7 g W g 7. The price for a round-trip bus ride from a universit to center cit is $., but it is possible to purchase a monthl commuter pass for $. with which each round-trip ride costs an additional $.. (a) Find equations for the cost C of making round-trips per month under both pament plans, and graph the equations for (treating C as a continuous function of, even though assumes onl integer values). (b) How man round-trips per month would a student have to make for the commuter pass to be worthwhile? 7. A student must decide between buing one of two used cars: car A for $ or car B for $. Car A gets miles per gallon of gas, and car B gets miles per gallon. The student estimates that gas will run $ per gallon. Both cars are in ecellent condition, so the student feels that repair costs should be negligible for the foreseeable future. How man miles would the student have to drive before car B becomes the better bu?